Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

The eigenvalue associated with each eigenvector quantifies the prominence of the contribution of this particular co-ordinate for explaining the variance of the data. The eigenvector with highest eigenvalue indicates the most important axis in the data space: the axis with highest projected variance. Visualization in this framework amounts to projecting the high-dimensional data (the 12-dimensional pitch class frequency space or, respectively, the 24-dimensional key frequency space) onto a small number (typically 2 or 3) of eigenvectors with high eigenvalues. Hereby only insignificant dimensions of the data space are discarded, leading, effectively, to a plot of high-dimensional data in 2d or 3d space.

In principal component analysis by rotating the co-ordinate system, the Euclidean distances between data points are preserved. Correspondence analysis is a generalization of principal component analysis: The $2 x$ distance (a generalization of the Euclidean distance) between data points is preserved.

If the data matrix is not singular and not even symmetric, generalized singular value decomposition instead of eigenvalue decomposition yields two sets of factors $u1,...,ud$ and $v1,...,vd$ instead of one set of eigenvectors. So either for the $m$ -dimensional column vectors of the data matrix the co-ordinate system can be rotated yielding a new co-ordinate system given by the column factors $u1,...,ud$ , or the $n$ -dimensional row vectors of the data matrix are expressed in terms of co-ordinates in the new co-ordinate system of row factors $v1,...,vd$ . In principal component analysis each eigenvector is associated with an eigenvalue. In the same sense for each pair of column and row vectors $uk$ and $vk$ , an associated singular value $dkk$ quantifies the amount of variance explained by these factors (see the appendix, Section 6 for technical details). Consider the conditional relative frequency of pitch classes $P| K F$ being the data matrix. If we project the 12-dimensional pitch class profile $P |K=i f$ into the space spanned by all $d$ vectors $v1,$ ..., $vd$ and represent each profile $P| K=i f$ by its $d$ -dimensional co-ordinate vector $si$ , then the $2 x -$ distance between $P|K=i f$ and $P|K=l f$ equals the Euclidian distance between the co-ordinate vectors $si$ and $sl$ of their projections. But if we only use the two co-ordinates with highest singular value, instead of all $d$ co-ordinates, then all distances are contracted and more or less distorted, depending on the singular values.

A biplot provides a simultaneous projection of features $K$ and $P$ into the same space. Both the co-ordinates of a $K$ -profile in the co-ordinate system of the $uk$ ’s and the co-ordinates of a $P$ -profile in the co-ordinate system of the $vk$ ’s are displayed in the same co-ordinate system. Such a biplot may reveal the inter-set relationships.

3 Circle of Fifths in the Keyscape

We will now investigate the set of Preludes & Fugues in Bach’s WTC. For each part of WTC there is a one-to-one mapping between all 24 pairs of Preludes & Fugues and all 24 Major and minor keys. The Table above shows how each key - that implies each Prelude & Fugue pair also - can be represented by a frequency profile of pitch classes. The pitch class frequency profiles can either contain the overall symbolic durations from the score or the accumulated cq-profiles from a performance of that piece. Correspondence analysis visualizes inter-key relations on keyscapes based on pitch class profiles. The projection of pitch classes homo-